Richard Wade: Duality groups and Cohen-Macaulay spaces

Abstract:

Via Poincaré duality, fundamental groups of aspherical manifolds have (appropriately shifted) isomorphisms between their homology and cohomology. In a 1973 Inventiones paper, Bieri and Eckmann defined a broader notion of a duality group, where the isomorphism between homology and cohomology can be twisted by an object called a dualizing module. Examples of these groups in geometric topology (after passing to a finite-index subgroup) include \(\mathrm{GL}(n, \mathbb{Z})\), mapping class groups, and automorphism groups of free groups.

Every example of a duality group that we know of has a classifying space satisfying a local condition called the Cohen-Macaulay property. Such spaces also satisfy weaker (twisted) versions of Poincaré duality via their local homology sheaves (or local cohomology cosheaves). Although we do not yet know how to directly link the topological and algebraic duality results, I will describe how this may provide more information about the cohomology of groups, such as \(\operatorname{Out}(F_N)\), that we already know satisfy Bieri-Eckmann duality.

Based on joint work with Thomas Wasserman.